Notations for points and vectors

The first thing that we need is to be able to define precisely the points that we want to put our objects at the required position.

Points, vectors and barycenters

A point p in space may be defined by the following expressions:

p=point(3,4,6)
p=origin+3*X+4*Y+6*Z
p=0.5*origin+0.5*point(6,8,12) # p is in the middle between origin and point(6,8,12)

A vector may defined by the following expressions:

v=Vector(3,4,6)
v=3*X+4*Y+6*Z # a linear combination of vector is a vector
v=0.5*(point(6,8,12)-origin) # a difference between 2 points is a vector

The barycenter is a math notion generalizing the center : It makes sense to take linear combinations of points as long as the sum of coefficients is 1: this is a barycenter. The following code defines a center, and the barycenters of the points p1,p2,p3,p4 with coefficients n1=.2,n2=.3,n3=..4,n4=.1 which makes sense since n1+n2+n3+n4==1

center=0.5*p1+0.5*p2
n1=.2,n2=.3,n3=..4,n4=.1
barycenter=n1*p1+n2*p2+n3*p3+n4*p4

Recall that in contrast, in maths it is not allowed to add 2 points : this makes no sense. In pycao, you can do it but we encourage you not to do it ( the output is a massic point of mass 2 in an abstract massic space, neither a point (mass 1) nor a vector (mass 0)). You can check by yourself

print(origin+origin)
>>Mass Point  [0 0 0 2]

Remark: in the notation, point(x,y,z), the coordinates (x,y,z) are global coordinates. This is not to be confused with the notation object.point(x,y,z) that we will explain later, and where coordiantes are relative to the object.

Reminder on points and Vectors

For those who have forgotten the maths, here are some rule of thumbs as a reminder for notions of points and vectors,

Usually, points and vectors are displayed picturally as follows.

Recall that we have the formulas :

point+vector=point
point-point=vector
vector+vector=vector

In the above picture, we have:

yellow point+blueVector=orangePoint
redPoint-orangePoint=GreenVector
blueVector+greenVector=redPoint-yellowPoint

This formalism of points and vectors is known to pycao.